# Perceptron and Decision Boundaries

The single-unit linear classifier and the hyperplane it induces — the historical starting point and the clearest illustration of linear separability's limits.

**Rests on:** Linear Regression, Two-Class and Multiclass Classifiers

**Leads to:** Multi-Layer Perceptrons

*Deep Learning · Deep Learning*


## Why learn this

You want to understand the absolute basics of how a machine draws a line to sort things into two piles — like separating good emails from spam, or sick patients from healthy ones. This is the oldest and simplest learning machine in AI: a single unit that adjusts a straight line (in two measurements) or a flat dividing surface (in three or more measurements) until it correctly splits the two kinds of things. Everything you will study later — neural networks, deep learning, LLMs, image recognition — is built by stacking many such units and adding twists to their outputs. Mastering this unit gives you the vocabulary and visual intuition for what 'learning' means: adjusting a dividing line until it separates the data. It also shows you the exact reason why a single unit fails (it cannot learn XOR, which is a specific pattern where items are grouped in two opposite corners of a square rather than on one side of a line), which is the historical motivation for deep networks with hidden layers. So this page is your foundation: learn it and you can name what every later model is doing under the hood.


## The intuition

Imagine you have a basket of apples and oranges, and you want a machine to tell them apart by just two measurements: weight and colour. You plot each fruit on a graph — weight on the horizontal axis, colour on the vertical. Apples cluster on one side, oranges on the other. The perceptron's job is to find a straight line that separates the two clusters. Once it finds that line, it can classify any new fruit: if it falls on the apple side of the line, call it an apple; if on the orange side, call it an orange.

The machine works by calculating a score for each fruit. It multiplies the weight by a number (how important weight is), adds the colour multiplied by another number, and then adds a constant offset — called the bias — which simply shifts the line left or right before the decision is made. If this total score is above zero, it says 'apple'; if below zero, 'orange'.

The key insight: the perceptron only learns the *position and tilt* of that line. It does not learn anything else. If the clusters are not separable by a straight line (like apples and oranges mixed in a circle), the perceptron will never get them all right, no matter how long you train it. That is its fundamental limit, and it is exactly what stalled AI research for a decade in the 1960s.


## An analogy

Think of a lighthouse keeper on a rocky coast. The keeper's job is to decide whether a ship is safe to approach the harbour (left side) or must be turned away (right side). The keeper watches two signals: the ship's distance from the rocks and its speed. Each ship is a point on a map — distance on the horizontal axis, speed on the vertical. The keeper has a straight line drawn on their chart, and they decide by which side of the line the ship falls. They start with a random line, but every day a harbour master tells them whether each ship was actually safe or not. When the keeper gets a ship wrong, they nudge the line — tilt it a bit, shift it left or right — to correct that mistake. Over weeks of corrections, the line settles where it separates safe ships from unsafe ones as well as possible.

That is exactly what the perceptron does: it adjusts its dividing surface each time it makes a classification error. The analogy breaks down in two ways. First, the keeper only has two inputs (distance and speed), but a real perceptron can have hundreds or thousands of inputs — you cannot draw a line in 10 dimensions, but the mathematics is exactly the same: a flat surface (a hyperplane, which is just the generalization of a flat plane to any number of dimensions) that divides space into two halves. Second, the keeper is given the correct answer by the harbour master, but in real machine learning you often do not have a perfect oracle — you have noisy labels. Still, the core idea — adjusting the boundary based on mistakes — is identical.


## Definition

A perceptron is a single-unit linear classifier that computes a weighted sum of its inputs plus a bias (a constant offset that shifts the decision line before classification), and if that sum is above zero it outputs one class, otherwise the other; the set of inputs that produce a sum of exactly zero forms a hyperplane (the multi-dimensional version of the boundary line) that separates the two classes.


## How it connects

You have just learned about functions — a rule that takes an input and gives an output. The perceptron is a very specific kind of function: it takes a list of numbers (the inputs) and returns either 0 or 1 (the class). It uses a weighted sum, which is arithmetic — you multiply each input by a number and add them up. You have also seen dot products in your notes (from the linear algebra topic). The perceptron's weighted sum is exactly a dot product between the input vector and the weight vector, plus a bias. That dot product measures how much the input 'aligns' with the weight direction. The decision boundary is the set of points where that dot product plus bias equals zero — a straight line in 2D, a plane in 3D, a hyperplane in higher dimensions. This connects to your notes on Activation Functions, because the perceptron uses a step activation (output 1 if positive, 0 otherwise), and to your notes on Gradient Descent, because training the perceptron is done by adjusting weights to minimize a loss. But note: the perceptron update rule is not full gradient descent — it is a simpler, mistake-driven rule that only nudges when it gets a classification wrong.


## The mathematics

The perceptron is a simple rule for sorting items into two groups, like apples and oranges, based on their features. It works by drawing a straight line (in two dimensions) or a flat plane (in three or more dimensions) through the data. Items on one side of the line are called class 1; items on the other side are class 0.

To draw this line, the perceptron uses two things: weights and a bias. The weights tell us how important each feature is. For example, if weight for 'weight in grams' is large, then the total weight matters more than the color score. The bias acts like a threshold or a shift, moving the line left or right regardless of the features.

Here is how the perceptron makes a decision for one item:

1. It takes the input features (e.g., x1 = 150 grams, x2 = 8 color score).
2. It multiplies each feature by its corresponding weight (w1 and w2). This is called the dot product. If you have weights w1 and w2, and inputs x1 and x2, the dot product is w1 times x1 plus w2 times x2.
3. It adds the bias (b) to that sum. The result is a single number, often called z.
4. If z is greater than zero, the perceptron predicts class 1. If z is less than or equal to zero, it predicts class 0.

The equation for this decision boundary—the line where the prediction switches from 0 to 1—is:
w1 * x1 + w2 * x2 + b = 0.

This equation defines a straight line in 2D space. Any point (x1, x2) that makes the left side positive is on one side of the line; any point that makes it negative is on the other.

Training the perceptron means adjusting w1, w2, and b so that the line correctly separates the known examples. The algorithm works like this:

1. Pick a training example with a known true class y (either 0 or 1).
2. Calculate z using the current weights and bias.
3. Make a prediction ŷ: if z > 0, ŷ = 1; otherwise ŷ = 0.
4. Check if the prediction is wrong. If ŷ does not equal y, the perceptron made an error.
5. If there was an error, update the weights and bias to reduce that error in the future. The update rule adds a small fraction of the input features to the weights, scaled by the learning rate (η) and the error (y - ŷ).

The specific updates are:
w1 becomes w1 + η * (y - ŷ) * x1
w2 becomes w2 + η * (y - ŷ) * x2
b becomes b + η * (y - ŷ)

If the prediction was correct (ŷ = y), then (y - ŷ) is zero, and nothing changes. If the prediction was wrong, (y - ŷ) is either +1 or -1, causing the weights to shift in a direction that makes the correct class more likely next time.

Note: This simple perceptron only works if the two classes can be separated by a single straight line. If the data is mixed up in a way that no straight line can separate them (like the XOR problem), this algorithm will never find a perfect solution and will keep updating forever.


## A worked example

Let us walk through one step of training with real numbers.

**Setup:**
- We have two features: x1 (weight in grams) and x2 (color score from 0 to 10).
- We want to separate Apples (class 1) from Oranges (class 0).
- Initial weights: w1 = 0.5, w2 = -0.2.
- Initial bias: b = 0.0.
- Learning rate (η): 0.1. This is a small number that controls how big each step is.

**Step 1: Process an Apple (True Class y = 1)**
- Input: x1 = 150, x2 = 8.
- Calculate z (the raw score):
  z = (w1 * x1) + (w2 * x2) + b
  z = (0.5 * 150) + (-0.2 * 8) + 0.0
  z = 75 + (-1.6) + 0
  z = 73.4
- Make prediction ŷ:
  Since z (73.4) is greater than 0, the perceptron predicts class 1.
- Check error:
  True class y = 1. Prediction ŷ = 1.
  Error (y - ŷ) = 1 - 1 = 0.
- Update weights:
  Since the error is 0, no update happens. The weights stay w1 = 0.5, w2 = -0.2, b = 0.0.

**Step 2: Process an Orange (True Class y = 0)**
- Input: x1 = 100, x2 = 2.
- Calculate z using the current weights:
  z = (w1 * x1) + (w2 * x2) + b
  z = (0.5 * 100) + (-0.2 * 2) + 0.0
  z = 50 + (-0.4) + 0
  z = 49.6
- Make prediction ŷ:
  Since z (49.6) is greater than 0, the perceptron predicts class 1 (Apple).
- Check error:
  True class y = 0. Prediction ŷ = 1.
  Error (y - ŷ) = 0 - 1 = -1.
  The perceptron made a mistake: it called an Orange an Apple.

**Step 3: Update weights to correct the error**
We use the update rule: new weight = old weight + learning_rate * error * input_feature.

- Update w1:
  w1_new = 0.5 + 0.1 * (-1) * 100
  w1_new = 0.5 - 10
  w1_new = -9.5

- Update w2:
  w2_new = -0.2 + 0.1 * (-1) * 2
  w2_new = -0.2 - 0.2
  w2_new = -0.4

- Update bias b:
  b_new = 0.0 + 0.1 * (-1)
  b_new = -0.1

**Result after this step:**
The new weights are w1 = -9.5, w2 = -0.4, and bias b = -0.1.
The new decision boundary equation is:
-9.5 * x1 - 0.4 * x2 - 0.1 = 0

Let us check if this helps the Orange example:
z_new = (-9.5 * 100) + (-0.4 * 2) + (-0.1)
z_new = -950 - 0.8 - 0.1 = -950.9
Since z_new is negative, the perceptron now correctly predicts class 0 (Orange) for this input.

**Note on magnitude:**
The weights changed dramatically because the input x1 (100) was large and the learning rate (0.1) multiplied by it created a big step (-10). In practice, we often normalize inputs or use a smaller learning rate to make the line move gradually rather than jumping wildly.


## From scratch, in Python

```python
# Perceptron from scratch in plain Python

# Step 1: Define the training data as a list of (x1, x2, y) tuples
# A tuple is like a list but cannot be changed. Each example is (features..., label).
training_data = [
    (2, 3, 1),   # example 0: x1=2, x2=3, class 1
    (1, 1, 0),   # example 1: x1=1, x2=1, class 0
    (3, 4, 1),   # example 2: x1=3, x2=4, class 1
    (0, 1, 0)    # example 3: x1=0, x2=1, class 0
]

# Step 2: Initialize weights and bias
w = [0.0, 0.0]
b = 0.0
learning_rate = 0.1

# Step 3: Define a function that computes the dot product of two lists
# We loop over indices, multiply each pair, and add to a running total.
def dot_product(a, b):
    total = 0
    for i in range(len(a)):
        total += a[i] * b[i]
    return total

# Step 4: Define the prediction function
def predict(x):
    z = dot_product(w, x) + b  # weighted sum plus bias
    if z > 0:
        return 1
    else:
        return 0

# Step 5: Training loop — repeat several epochs (passes over the data)
for epoch in range(10):
    print(f"Epoch {epoch}:")
    for x1, x2, y in training_data:
        x = [x1, x2]
        y_pred = predict(x)
        error = y - y_pred
        # If error is zero, no update
        if error != 0:
            # Update each weight: w_i = w_i + learning_rate * error * x_i
            w[0] += learning_rate * error * x[0]
            w[1] += learning_rate * error * x[1]
            # Update bias: b = b + learning_rate * error
            b += learning_rate * error
            print(f"  Updated weights to {w}, bias to {b} for example ({x1},{x2},{y})")
    # After each epoch, print the current line
    print(f"  Current decision line: {w[0]}*x1 + {w[1]}*x2 + ({b}) = 0")

# Step 6: Test the trained perceptron on a new point
new_x = [2.5, 2.0]
print(f"Prediction for {new_x}: {predict(new_x)}")

```


## The same thing, with a framework

```python
You could use NumPy to vectorize the operations: `w = np.array([0.0, 0.0])`, `x = np.array([x1, x2])`, and the dot product becomes `np.dot(w, x)`. The update rule becomes `w += learning_rate * error * x`. NumPy does the same multiply-and-add loop underneath, but faster for large arrays. For a production model you would use a library like scikit-learn's `Perceptron`, which handles convergence criteria and scaling automatically — but under the hood it implements the exact loop you just wrote.
```


## What the code does

The code is deliberately verbose to show every step. First we define the training data as a list of tuples. A tuple is like a list but cannot be changed after you create it — we use it to keep each example's values (x1, x2, label) together. Then we initialize the weights to zero and the bias to zero, and set a learning rate of 0.1 (a small number so changes are gradual). The `dot_product` function is the heart of the perceptron: it takes two lists, loops over them with an index `i`, multiplies the i-th elements together, and adds them to a running total — that is the weighted sum. The `predict` function adds the bias to the dot product and returns 1 if positive, otherwise 0. The training loop runs 10 epochs (complete passes over the data). For each example, it computes the prediction, then the error (true class minus prediction, so 0 if correct, +1 or -1 if wrong). If the error is non-zero, it updates every weight by adding `learning_rate * error * input_value`, and the bias by adding `learning_rate * error`. After each epoch, it prints the current decision line. Finally, it tests on a new point to show the model works.


## Common misconceptions

**The perceptron learns any pattern** — If I train it long enough, the perceptron will eventually classify any data correctly, no matter how complex.

The perceptron can only draw a straight line (or a hyperplane in higher dimensions). If the data is not linearly separable — like the two classes in XOR where points alternate in a checkerboard pattern — no straight line can separate them. The training algorithm will keep jumping and never converge. In fact: The perceptron can only learn linearly separable patterns. For anything more complex, you need multiple layers with nonlinear activations (the perceptron's successor, the multi-layer perceptron).

**Weights are probabilities or percentages** — Weights must be between 0 and 1, and they should sum to 1, like a weighted average.

Weights are just numbers, and they can be any real value — positive, negative, large, or small. They are not normalised. A negative weight means that input pushes the decision toward the other class. There is no constraint that they must sum to anything. In fact: Weights are free parameters, learned from data. They represent the direction and magnitude of each feature's influence on the classification. They can be any real number.

**The decision boundary is a circle or a curve** — The perceptron can draw a curved boundary if the data is curved.

The perceptron's boundary is always a straight line (in 2D) or a flat hyperplane (in higher dimensions). There is no curve. If the data needs a curve, the perceptron will fail. In fact: The decision boundary is always linear. Curved boundaries require nonlinear transformations (e.g., kernel methods) or deeper networks with nonlinear activations.

**The perceptron output is a probability** — The perceptron outputs a number between 0 and 1 that I can interpret as a probability of belonging to class 1.

The perceptron outputs a 0 or a 1 based on a threshold (zero). The raw score before the threshold can be any real number, but the output is a hard class label, not a probability. For probabilities you need a logistic regression (sigmoid activation), which is a different model. In fact: The perceptron gives a hard classification (0 or 1). If you need a probability, use a logistic regression, which applies a sigmoid to the raw score to produce a value between 0 and 1.

**Bias is a bad thing (as in unfairness)** — The bias term in the perceptron is the same as bias in the sense of discrimination or unfairness.

In machine learning, 'bias' has a technical meaning: it is a constant added to the weighted sum that shifts the decision boundary. It is not about social bias. It allows the boundary to not always pass through the origin. In fact: The bias is a learned parameter that translates the decision boundary. It is unrelated to bias as unfairness, though models can learn unfair biases from data — that is a separate issue.


## In short

A perceptron is the simplest learning machine: it computes a weighted sum of inputs plus a bias, applies a step at zero, and adjusts its weights based on errors. Its decision boundary is always a flat hyperplane, which is its strength (simplicity) and its weakness (cannot learn XOR or anything non-linear). It is the foundational unit of all neural networks — stack perceptrons with nonlinear activations and you get deep learning. Remember: the perceptron learns a line; deep learning learns a very complex surface composed of many such lines.


## Where this is used

Where does the perceptron show up in modern AI? Almost everywhere as a building block, but rarely alone. In **large language models (LLMs)** like GPT-4, the very first processing step is a linear transformation: each token's embedding vector is multiplied by a weight matrix (a set of perceptrons, one per output dimension) plus a bias. That is a perceptron in a matrix form. The transformer's feed-forward layers are also stacks of perceptrons with nonlinear activations. The decision boundary idea persists: each neuron in a layer is drawing a hyperplane in the feature space, deciding what that feature 'fires' on. In **small language models that run on phones**, the same linear layers appear, but they are compressed to save memory — quantisation reduces the precision of the weights, which changes the decision boundary slightly. A developer using a framework like PyTorch writes `nn.Linear(in_features, out_features)`, which is a whole matrix of perceptrons — each row is one unit's weights. The training loop, however, is the same concept: gradient descent adjusts the weights to move the decision boundaries to fit the data. In **retrieval-augmented generation (RAG)**, the embedding model that converts text to vectors is often a neural network built from linear layers and activations. The similarity between a query and a document is often a dot product — the same operation the perceptron uses. A developer might use a pre-trained embedding model and then do a dot product with each document vector; the highest score wins. The perceptron's decision boundary is a simple version of that: instead of a hard line, you use a threshold on the similarity score to decide whether to retrieve a document. In **agentic systems that plan and call tools**, the planning module might use a classifier to decide whether the current context is sufficient to call a tool or needs more user input. That classifier is often a small neural network — a stack of perceptrons. The developer would train it on labelled dialogues, and the training loop is the same perceptron update, just with more layers and backpropagation. In **everyday applied machine learning** — recommendation systems, spam filters, fraud detection — linear models (logistic regression, support vector machines) are built on the same foundation. A bank's fraud detection model might flag a transaction if a weighted sum of features (amount, location, time) crosses a threshold. That is a perceptron. The developer might use scikit-learn's `LogisticRegression`, which trains the same weights but with a more robust loss function. The perceptron's historical limitation — it cannot learn XOR — motivated the invention of multi-layer networks with nonlinearities. That is why every modern model has an activation function like ReLU after each linear layer. Without that nonlinearity, stacking perceptrons would collapse into a single linear classifier, and the network could never learn anything more complex than a straight line. So the perceptron is not just a historical curiosity; its failure is the reason deep learning works.


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